Workshop on degenerations of Calabi-Yau manifolds (IHP, Paris, May 14-16, 2018)
Valentino Tosatti, Chenyang Xu and Sébastien Boucksom are coorganizing
a workshop on degenerations of Calabi-Yau manifolds at the Institut
Henri Poincaré (Paris), from May 14 to 16, 2018.
Program: the workshop consists in three-hour mini-courses and one-hour talks.
Sébastien Boucksom: Calabi-Yau degenerations and hybrid spaces.
Valentino Tosatti: Adiabatic limits of Ricci-flat Kähler metrics.
Chenyang Xu: Essential skeleta and dlt models.
Ruadhaí Dervan: Canonical metrics on fibrations and maps in the adiabatic limit
Yang Li: Collapsing Calabi-Yau metrics on Lefschetz K3 fibred 3-folds
Yuchen Liu: The normalized volume of a singularity is lower semi-continuous
Tony Yue Yu: The Frobenius structure conjecture in dimension two
Yuguang Zhang: Equivalences for degenerations of Calabi-Yau manifolds
Practical information and Schedule:
The meeting will take place in the room 314 of the Institut Henri Poincaré
|Monday, May 14
||Tuesday, May 15
||Wednesday, May 16
10.00am - 10.45am
10.45am - 11.15am
11.15am - 12.00pm
12.00pm - 1.30pm
1.30pm - 2.15pm
2.15pm - 2.30pm
2.30pm - 3.30pm
3.30pm - 4.00pm
4.00pm - 5.00pm
References and abstracts
S.Boucksom, M.Jonsson: Tropical and non-Archimedean limits of degenerating families of volume forms.
T.de Fernex, J.Kollár, C.Xu: The dual complex of singularities
M.Gross, V.Tosatti, Y.Zhang: Collapsing of abelian fibred Calabi-Yau manifolds
J.Kollár, C.Xu: The dual complex of Calabi-Yau pairs
J.Nicaise, C.Xu: The essential skeleton of a degeneration of algebraic varieties
J.Nicaise, C.Xu, T.Yue Yu: The non-archimedean SYZ fibration
V.Tosatti: Adiabatic limits of Calabi-Yau metrics.
V.Tosatti, Y.Zhang: Collapsing hyperkähler manifolds
Ruadhaí Dervan: A
natural question is when a Kähler manifold admits a canonical choice of
Kähler metric. The sort of canonical metrics I will discuss are a
generalisation of Kähler Einstein metrics. I will discuss some
existence and non-existence results in the situation when the Kähler
manifold is fibred over a lower dimensional manifold. It is worth
remarking that this is a sort of inverse to Tosatti's mini-course; we
attempt to understand the existence of canonical metrics on fibred
manifolds through properties of the base, rather than understanding
limiting behaviour of families of canonical metrics on fibred
(Calabi-Yau) manifolds, which then induce certain metrics on the base.
This is joint work with Julius Ross and Lars Sektnan.
Yang Li: I will discuss the problem of describing the collapsing CY
metrics on a CY 3-fold with a Lefschetz K3 fibration. Collapsing CY
metrics is a well studied subject, but most of the previous works
concentrate on the behaviour away from the singular fibres, and the
full description of the metric was only available in a very small
number of cases, mostly relying on very favourable gluing ansatz.
The main geometric insight is that at the scale where the fibres
have volume 1, the neighbourhood of the nodal fibre has local
non-collapsing behaviour, and converges in the pointed Gromov Hausdorff
sense to nodal K3 times C. Furthermore, there is a much finer scale
near the nodal points in the fibration, where the scaled limit is a CY
metric on C3 with maximal volume growth and singular tangent cone at
infinity. This model metric was previously constructed by the author in
a separate work. The difficulty of the gluing lies in the coarse nature
of the gluing ansatz, and the fact that the metric has many types of
characteristic behaviours at different scales. We overcome this by
developing a sharp linear theory, using some earlier ideas of Gábor
Yuchen Liu: Motivated by work in differential geometry, Chi Li
introduced the normalized volume of a klt singularity as the minimum
normalized volume of all valuations centered at the singularity. This
invariant carries some interesting geometric/topological information of
the singularity. In this talk, we show that in a Q-Gorenstein flat
family of klt singularities, normalized volumes are lower
semicontinuous with respect to the Zariski topology. As an application,
we show that K-semistability is a very generic or empty property in a
Q-Gorenstein flat family of Q-Fano varieties. This is a joint work with
Tony Yue Yu: The Frobenius structure conjecture is a conjecture about the geometry
of rational curves in log Calabi-Yau varieties proposed by
Gross-Hacking- Keel. It was motivated by the study of mirror symmetry.
It predicts that the enumeration of rational curves in a log Calabi-Yau
variety gives rise naturally to a Frobenius algebra satisfying nice
properties. In a joint work with S. Keel, we prove the conjecture in
dimension two. Our method is based on the enumeration of
non-archimedean holomorphic curves developed in my thesis. We construct
the structure constants of the Frobenius algebra directly from counting
non-archimedean holomorphic disks. If time permits, I will also talk
about compactification and extension of the algebra.
Yuguang Zhang: In this talk, we study the relationships among
Ricci-flat Kähler-Einstein metrics, cohomology classes of holomorphic
volume forms, and the Weil-Peterson metric of degenerations of